Int. J. Engng Ed. Vol. 22, No. 5, pp. 1070±1076, 2006
Printed in Great Britain.
0949-149X/91 $3.00+0.00
# 2006 TEMPUS Publications.
Comparisons Between Performances in a
Statics Concept Inventory and Course
Examinations*
PAUL S. STEIF
Department of Mechanical Engineering, Carnegie Mellon University, Pittsburgh, PA 15213, USA.
Email: steif@andrew.cmu.edu
MARY HANSEN
Department of Secondary Education and Graduate Studies, Robert Morris University, Pittsburgh,
PA 15213, USA. Email: hansen@rmu.edu
Multiple analyses of results are presented from the Statics Concept Inventory, a multiple choice
test assessing conceptual knowledge in engineering statics. Results are based on the administration
of this test to 1331 students in ten classes at seven US universities during the 2004±2005 academic
year. Evidence confirming the validity and reliability of the test is offered. Detailed comparisons
are made between inventory scores and performance on course examinations, including evaluations
of how inventory sub-scores on specific concepts correlate with performance on certain types of
problems. Finally, based on analysis of the prevalence of various answer choices, common
misconceptions are identified.
Keywords: statics; assessment; multiple choice test; test administration
summative assessment of conceptual gains.
However, there is also the potential for using the
inventory as a means of formative assessment,
particularly since the analysis of the test results
provides scores on distinct concepts. Consider, for
example, the inventory to be administered a few
weeks prior to the end of Statics, or at the
beginning of a follow-on course (e.g., Dynamics
or Mechanics of Materials). Then, the performance of individual students on different concepts
could be fed back in real time to enable additional
instruction that is tailored to the specific needs of
individual students.
The current paper presents results for the
2004±2005 version of the inventory, which has
been administered at a number of institutions. In
addition, through detailed comparisons of inventory results with performances on course examinations, evidence is offered for the potential value of
such concept-specific feedback.
INTRODUCTION
ASSESSMENT IS increasingly recognized as
crucial to improving learning [1, 2]. Since conceptual understanding is strongly tied to the ability to
transfer newly acquired knowledge [3]Ða major
goal of many engineering science coursesÐassessment ought to include conceptual knowledge. One
approach, pioneered with the Force Concept
Inventory [4], has been to assess conceptual understanding through multiple choice tests in which
questions and the various answer choices are
carefully constructed to measure knowledge of
important concepts and identify common misconceptions. Concept inventories have recently been
under development for a number of engineering
subjects [5].
One such inventory, the Statics Concept Inventory (henceforth referred to here as `the inventory'), has been developed to assess conceptual
understanding for the engineering course Statics
[6]. Underlying the test was an extensive observational analysis of the products of novice problem
solving and a formulation of those observations
into a conceptual framework [7]. The inventory
comprises 27 questions which focus on eight
concepts in Statics.
Like most concept inventories, this test is
intended to be administered towards the beginning
and towards the end of a Statics course, offering a
RESULTS FOR 2004±2005
ADMINISTRATIONS
This paper reports on data from administrations
of the inventory during the 2004±2005 academic
year at US universities. In all cases, the test was
administered near or after the completion of
Statics (post-test) to a total of 1331 students in
10 different classes (see Table 1). (The test was
administered at two additional schools; one
lost the data and a second mistakenly used the
* Accepted 20 November 2005.
1070
Performances in a Statics Concept Inventory and Course Examinations
Table 1. Classes taking inventory (test methodÐpaper and
pencil: 1, 4, 5, 7, 10; electronic: 2, 3, 6, 8, 9)
Class
1
2
3
4
5
6
7
8
9
10
N
Course
University
97
38
35
42
419
262
158
75
42
163
Statics
Statics
Statics
Statics
Statics
Mech. of Mat.
Statics
Mech. of Mat.
Mech. of Mat.
Statics
A
B
B
C
D
D
E
E
F
G
2003±2004 test.) As reported below, the inventory
was also administered as a pre-test (prior to
Statics) in six of these classes.
In some cases, the test was administered in class
with pencil and paper; in other cases, students took
the test electronically. (In one class, students
entered answers into a spreadsheet that was
emailed in.) Students were aware that their score
on the inventory would not affect their class grade,
although at some institutions points were given for
merely completing the test. Time limits of 50 to 60
minutes were imposed for tests taken in class; no
limit was imposed when the test was completed
electronically. While students were told to
complete the test on their own, there was no
monitoring of the students who took the test
electronically. As will be seen from the data
below, no noticeable differences in performance
were found between different methods of test
administration.
Classes 2 and 3 were taught by the same
instructor in two successive semesters of Statics.
Classes 5 and 6 correspond to Statics and
Mechanics of Materials, respectively, taught
during successive semesters at the same university;
the Mechanics of Materials course consisted
mostly of students from the prior Statics course.
Classes 7 and 8 likewise corresponded to Statics
and Mechanics of Materials courses. With the
exception of Class 1, which is a private research
university (CMU), the remaining classes were
at public universities, all with graduate and
undergraduate programs with various rankings.
Means and standard deviations for the post-test
scores, and the mean for the pre-test scores (where
administered), are shown in Table 2 for each class.
In the transition from a paper-based to web-based
administration, personal data were not collected
for many students. Therefore, the authors cannot
report reliably on any differences of performance
between genders and ethnic groups.
With the exception of Class 1, all schools had
highly comparable pre-test scores. Students in
Class 4 were instructed to leave blank any questions that they did not know how to answer.
Considering the fraction of questions that were
left blank, the pre-test scores in Class 4 would
likely have been comparable to other schools. It
can be seen that inventory scores were generally
1071
Table 2. Means and standard deviations for post-test, means
for pre-test, and normalized gains
Class
1
2
3
4
5
6
7
8
9
10
Post-Test
S.D.
Pre-Test
Norm.
Gain
20.51
13.87
14.17
11.62
12.14
12.08
13.64
14.37
12.24
13.04
4.24
4.66
4.12
4.14
4.59
4.92
5.14
5.09
4.10
4.96
10.46
7.04
7.13
4.82
7.35
*
7.41
*
*
*
0.61
0.34
0.35
0.31
0.24
*
0.32
*
*
*
similar for classes taking the paper and pencil and
electronic formats of the inventory.
The variation in the post-test scores for different
classes is more significant. Normalized gain was
used by Hake to compare pre-post differences in
performances on the Force Concept Inventory
among classes with very different starting points
[8]. Normalized gain is defined as the improvement
divided by the maximum possible improvement.
Clearly, the use of normalized gain is less important here, where classes have similar pre-test
scores.
Class 1 had a three-week introduction to Statics
in the prior year as part of a freshman Fundamentals of Mechanical Engineering course. The
relatively high post-test scores in Class 1 should
also be viewed in light of the fact that the instructor is also the main developer of the inventory (first
author). As was the case for all other classes,
students in Class 1 did not see the inventory
questions during the course of the semester. Nevertheless, it is likely that instruction in this class was
influenced by the developer/instructor's views of
Statics. Still, the higher pre-test scores for this class
may indicate a greater readiness for the subject
that allowed for greater gains.
A more extensive discussion of the evidence
supporting the reliability and validity of the
2003±2004 Statics Concept Inventory scores has
been presented [6]. For the current version of the
inventory, Cronbach's coefficient alpha, a measure
of the internal consistency reliability, was found to
be 0.82. This indicates that the inventory items
consistently measure the same underlying content.
One measure of the quality of an individual test
question is its ability to discriminate between
students. A question is more likely to be deemed
poor if students who answered it correctly
performed worse generally on other questions, as
compared with students who answered it incorrectly. To determine the extent to which individual
questions discriminate between students, the
sample was divided into students who scored in
the upper 27% overall and those who scored in the
lower 27% overall. The fraction of students in these
two groups that answered a given question
correctly was then computed. The Discrimination
Index for that question is defined as the difference
1072
P. S. Steif and M. Hansen
Table 4. Correlations between inventory and 1st, 2nd and
final exams in Mechanics of Materials classes
Class
r
8
0.65
9
0.57
0.39
0.46
0.19
0.30
Table 5. Correlations between course examinations, which are
found to be comparable to correlations between inventory
and examinations
Class
Fig. 1. Numbers of questions with Discrimination Index in
various ranges.
between these fractions. The numbers of questions
with Discrimination Index in various ranges are
shown in Fig. 1. Questions with Discrimination
Index above 0.4 are considered very good discriminators, between 0.3 and 0.4 are considered to be
good discriminators, and below 0.3 to be marginal
or poor discriminators [9]. Most items on this test
discriminate quite well.
RELATION TO CLASS EXAMINATIONS
Ultimately, instructors are concerned that
students perform well in Statics courses, and also
in subsequent courses. Thus, the inventory will be
a more valuable tool if it is indicative of performance on course examinations. Inventory scores of
students were correlated with their examination
scores (when they were available) using the Pearson Correlation coefficient r. For four Statics
courses, the comparison was based on the final
examination (Table 3); in Class 1, there was no
final exam, so the correlation was based on the
average of all four exams spread throughout the
semester. For the two Mechanics of Materials
courses, the inventory was compared with each
of three course examinations (Table 4).
Given that the inventory asks questions on
highly isolated and idealized aspects of statics, we
argue that these correlations are quite meaningful.
As a point of comparison, consider the correlations between exams within each class (Table 5).
For example, in Class 1, the correlations 0.65, 0.66,
0.63, 0.59, 0.63, and 0.71 are between exams 1 and
2, 1 and 3, 1 and 4, 2 and 3, 2 and 4, and 3 and 4,
respectively. With four exams, there are six pairwise correlations; with three exams, there are three
pair-wise correlations. By comparing the correlations provided in Tables 3, 4 and 5, it can be seen
that the correlations between the inventory and
Table 3. Correlations between inventory and final exams in
Statics classes
Class
r
1
2
3
7
10
0.62
0.59
0.24
0.48
0.41
1
2
3
7
8
9
Correlations between course examinations
0.65,
0.57,
0.42,
0.32,
0.60,
0.54,
0.66,
0.34,
0.33,
0.44,
0.60,
0.69,
0.63,
0.42,
0.66,
0.49,
0.70
0.64
0.59.
0.55.
0.13.
0.48.
0.63,
0.71,
0.25,
0.48,
0.71
0.73
0.47
0.59
class exams are generally of the same order as
correlations between class exams within each class.
CONCEPT-SPECIFIC SUB-SCORES
Feedback on students' performances that is more
specific will be of greater value to instructors than
just the overall scores. The questions in the inventory were each devised to correspond with one of
several concepts in Statics, as follows:
. Separating bodies (FBD): Identifying forces
acting on a subset of a system of bodies
. Equilibrium (Equil.): Consideration of both
force and moment balance in equilibrium
. Friction: trade-off between implications of
equilibrium and Coulomb's law
. Static equivalence (St.Eq.): Static equivalence
between forces, couples and combinations
. Roller: Direction of force between a roller and
the rolled surface
. Negligible friction (Neg.Fr.): Direction of force
between frictionless bodies in point contact
. Slot: Direction of force between a pin and the
slot of a member
. Representation (Repres.): Representing unknown loads at connections.
Hence, one should be able to infer from scores
information regarding specific concepts and
provide it to instructors. To investigate the feasibility and usefulness of such inferences, we have
undertaken: (1) to determine if independent subscores can be extracted consistently from the
inventory; and (2) if concept-specific sub-scores
are correlated with concept-specific performance
in class examinations.
The psychometric analysis regarding sub-scores
has included extensive factor analysis (both
exploratory and confirmatory). Details of this
analysis will be presented elsewhere. In short,
exploratory factor analysis provided evidence
that sub-scales arise from the data, and confirmatory factor analysis showed that these sub-scales
Performances in a Statics Concept Inventory and Course Examinations
Fig. 2. Statics examination problem (Class 1) with multiple
connected bodies (dimensions removed). The motor exerts a
given torque on the arm, which leads the ram to crush the object
with a force to be determined. Problem analyzed for errors in
forces at roller and slot (usual assumptions regarding frictionless pins apply).
are associated with the distinct concepts on which
the test questions are based. In the following
section we offer comparisons between performance
on course examinations and specific concept subscores from the inventory.
EXAM ERRORS AND CONCEPT
SUB-SCORES
Given that statistical analysis legitimized the
extraction of concept specific sub-scores from the
data, how are sub-scores related to performance in
course examinations? Two examination problems
from Class 1, depicted in Figs 2 and 3, were used to
study these relations in detail. From the problem in
Fig. 2, we assessed the ability of students to
recognize the directions of force exerted by a
roller on a surface and by a pin on a slot. From
the problem in Fig. 3, we assessed the ability of
students to allow the friction force to be less than
1073
N, if equilibrium is satisfied and no slippage
occurs.
Students in Class 1 were divided into two
groups: those who erred and and those who did
not err in recognizing the direction of the force of
the roller in the exam problem of Fig. 2. Students
in Class 1 were likewise split into those who did
and did not err in the force of the pin on the slot
(Fig. 2) and in the friction force (Fig. 3). For all
three splittings of the class, students who erred had
on average lower overall inventory scores and
lower sub-scores on all concepts. However, for
some concepts, the differences between the subscores of those who erred and those who did not
err were not statistically significant.
On the other hand, one expects the sub-score
Roller, for example, to be significantly different for
the two groups, those who erred and those who did
not err, on the roller force in the examination
problem. For each of the other exam errors,
there is one concept sub-score (Slot and Friction)
that also ought to discriminate most strongly
between those who erred and those who did not.
Based on their content, one expects the remaining
concepts to be less directly related to the three
exam errors.
In Table 6, we display the mean inventory subscores on these three concepts for the groups of
students that did and did not make these three
exam errors. In addition, a t-test was performed,
and the p-value for the test that the two groups
have the same mean concept sub-score is shown.
The smallest p-values are for concept sub-scores
that correspond to the concept at issue in the exam
problem. This indicates that the groups that erred
and did not err in the use of a concept in a class
exam differed significantly on those inventory
questions specifically addressing that concept. In
fact, with one exception (Roller sub-score on the
slot error), other differences between the groups
were not statistically significant (p > 0.05).
The above exam-inventory comparison was
based on results from Class 1 (students taught by
the primary inventory developer). Given the correlations between class examinations and overall
inventory scores shown above for other universities, it is likely that similar comparisons could be
made for exam problems at other schools. One
Table 6. Comparison of concept sub-scores for students who
made and did not make specific errors in course
examinations. Most significant differences between groups
were for concept most directly related to error
Roller:
47/97 erred
Slot:
33/97 erred
Fig. 3. Statics examination problem (Class 1) with friction.
Levels of force P to cause sliding of the blocks to be determined
for different levels of friction coefficient 2. Problem analyzed
for error of automatically associating the friction force with N.
Friction:
53/97 erred
Error
No error
p
Error
No error
p
Error
No error
p
Roller
Slot
Friction
0.645
0.867
0.002
0.646
0.818
0.023
0.723
0.803
0.265
0.872
0.927
0.199
0.778
0.964
0.001
0.868
0.939
0.075
0.652
0.760
0.127
0.667
0.729
0.42
0.610
0.826
0.001
1074
P. S. Steif and M. Hansen
Fig. 4. Statics examination problem from Class 2 compared
with inventory results. Students were asked to draw the shear
force and bending moment diagrams.
such analysis was conducted on a problem on the
final exam for Class 2 (Fig. 4).
Grading in this problem was done by the
instructor of Class 2 prior to, and independent
of, comparisons with the inventory. Scores for this
problem were based primarily on a student's ability to draw free body diagrams and write down
equations of equilibrium, all of which lead to shear
force and bending moment diagrams. Students
could earn a maximum of 6 points, and one
point was taken off for very minor errors. Thus,
we compared inventory scores of two groups of
students: those who earned 5 or 6 and those who
earned less than 5, indicating more substantial
errors in free body diagrams or equilibrium conditions (Table 7). Sub-scores on all concepts are
shown, along with (i) the difference in the means
divided by the standard deviation of the sub-scores
for the full group, and (ii) the p-value from a t-test
comparing the means of the two groups.
One should note the concepts that most discriminate between students who did well (5±6) on this
exam problem and those who did not (0±4). Subscores on concepts that are irrelevant to this
problem, namely Friction, Neglect of Friction,
Slot, and Representing Forces at Connections,
display non-significant differences. (The concept
Roller is also irrelevant to this problem; with
p ˆ 0.045, its difference is marginally significant.)
On the other hand, the most relevant concepts,
Free Body Diagrams, Equilibrium, and Static
Equivalence, display the most significant differences. The relevance of these concepts to this
problem is noteworthy. Certainly, the inventory
Fig. 5. Question requesting student to draw the free body
diagram of a subset of a system.
was devised with a focus on the concepts necessary
to solve problems of multiple inter-connected
bodies [6], such as that of Fig. 2. It is clear,
however, that the inventory is also quite relevant
to problems that instructors would view as very
distinct, namely, the drawing of shear force and
bending moment diagrams, as demanded by the
problem in Fig. 4.
COMMON MISCONCEPTIONS IN STATICS
One important use for the inventory is to assess
the prevalence of various types of errors. These can
point to widely held misconceptions that ought to
be addressed in instruction. Here we point out
errors that occurred most frequently.
An example of a question in the group `FBD' is
shown in Fig. 5. The student is asked to choose the
correct free body diagram for the collection of
blocks 1, 3 and 6, and the cords connecting
them. The most commonly chosen wrong answer
was the distractor with an `internal force', that is, a
force acting between two of the bodies in the
diagram (Tc in Fig. 6). This incorrect answer was
chosen by 15% of students, while the correct
answer was chosen by 66% of students. (In other
questions, the internal force distractor was chosen
by 26% of students.)
Table 7. Inventory performances of students in Class 2 with
at most minor errors (score 5 or 6, N ˆ 19) on problem in
Fig. 4, compared with students with more significant errors
(score 4 or less, N ˆ 19).
FBD
Equil.
Friction
St. Eq.
Roller
Neg. Fr.
Slot
Repres.
Total
Score 5±6
Score 0±4
Diff/SD
p
0.80
0.58
0.33
0.33
0.82
0.46
0.77
0.67
16.47
0.55
0.34
0.18
0.12
0.61
0.32
0.67
0.49
11.26
0.80
0.90
0.56
0.78
0.65
0.53
0.36
0.52
1.12
0.013
0.004
0.087
0.015
0.045
0.102
0.278
0.109
<0.0005
Fig. 6. Commonly selected wrong answer to question in Fig. 5,
featuring an internal force.
Performances in a Statics Concept Inventory and Course Examinations
Fig. 7. Question requesting student to choose loading that is
statically equivalent (to the 20 N m couple).
Fig. 8. Commonly chosen answer to question in Fig. 7 (force
taken to be statically equivalent to couple).
Fig. 9. Question requesting student to choose direction of force
on surface contacted by roller.
Fig. 10. Question requesting student to choose magnitude of
force between bodies in frictional contact.
Fig. 11. (a) Question in which student is requested to find load
exerted by hand at right which balances bar. (b) Most commonly chosen wrong answer.
Fig. 12. Question in which student is asked if loadings (I) and/
or (II) could be in equilibrium.
1075
An example of a question in the group `Static
Equivalence' is shown in Fig. 7. Students are given
one loading and asked to find a second loading
that is equivalent. The most commonly chosen
wrong answer was the distractor (Fig. 8) in which
the couple is taken to be equivalent to a force that
apparently produces the same moment. This incorrect answer was chosen by 26% of students, while
the correct answer was chosen by 32% of students.
For one question in the group `Static Equivalence',
distractors of this type were chosen more
frequently than the correct answer.
An example of a question in the group `Roller' is
shown in Fig. 9. Students are shown a system and
asked for the direction of the force of the roller on
the body on which it rolls (the usual assumptions
of a frictionless pin and so forth are given). The
most commonly chosen wrong answer was the
distractor in which the force is acting parallel to
the arm to which the roller is pinned, rather than
perpendicularly to the rolled surface. This incorrect answer was chosen by 20% of students, while
the correct answer was chosen by 58% of students.
In general, there is a strong tendency to assume
that forces always act parallel to elongated
members (as they do in a two-force member).
An example of a question in the group `Friction'
is shown in Fig. 10. In this question, students are
asked for the upward force exerted by the left
block on a center block. The choices are all
numbers. The distractor of 8 N was chosen by
39% of students, while the correct answer of 3 N
was chosen by only 32% of students. Students are
powerfully drawn to assume that the friction force
equals the coefficient of friction times the normal
force, rather than choosing a lesser force that
maintains equilibrium.
An example of question in the group `Equilibrium' is shown in Fig. 11. Students are shown the
bar (a) with the force applied and asked for the
loading exerted by a hand that grips the right end
and maintains the bar in equilibrium. The most
commonly chosen wrong answer is the distractor
in which a force and a moment try to balance each
other. The incorrect answer, shown as (b) in
Fig. 11, was chosen by 25% of students, while the
correct answer was chosen by 58% of students.
This parallels the error in static equivalency.
A second example of a question in the group
`Equilibrium' is shown in Fig. 12. The student is
asked whether one or both of the loadings could be
in equilibrium (all forces and couple have positive
magnitudes and act in the direction shown, but the
magnitudes are adjustable). Very few students
(11%) recognized that neither loading could
produce equilibrium, (I) because forces cannot
balance and (II) because moments cannot balance.
In particular, 70% of students accepted that (II)
could be in equilibrium, while 53% of students
accepted that (I) could be in equilibrium.
Students exhibit a wide range of misconceptions,
as evidenced by the choice of wrong answers;
however, certain errors stand out. Many students
1076
P. S. Steif and M. Hansen
have trouble rejecting internal forces, perhaps
revealing a lack of clarity regarding what body
exerts each force or which bodies' forces ought
legitimately to be included in a free body diagram.
Students often fail to grasp the necessity of independently satisfying force and moment summation and/
or that a couple carries no net force. Forces at various
connections sometimes have their directions set by
the nature of the connection, whereas students are
distracted from this by the shape of the body or by
other applied forces. Finally, the quantity N is only
the limit on the friction force; students are too often
convinced that N must be the actual level of the
friction force, even when there is no slip and a
lesser force is sufficient for equilibrium.
SUMMARY AND CONCLUSIONS
The Statics Concept Inventory is a multiple
choice test that assesses conceptual knowledge of
students in Statics. The test consists of 27 questions that capture eight distinct concepts and
include distractors (wrong answers) that have
been constructed based on observations of student
work. This paper is based on results from the
administration of this test to 1331 engineering
students in ten classes from seven US universities
during the 2004±2005 academic year. Previous
observations as to the psychometric soundness of
the test, including reliability and quality of items in
terms of discrimination indices have been reconfirmed. The bulk of the paper addresses comparisons between performance on the inventory and
performance on class examinations. It has been
found that there are significant correlations
between overall inventory scores and examination
scores, correlations that are on the order of correlations between class examinations. In addition, we
have shown that inventory sub-scores on specific
concepts can offer insight into the propensity to
make the analogous conceptual errors in examinations. Finally, by investigating the distribution of
wrong answers, we have identified common
misconceptions, information that may be valuable
for instruction. The inventory is available for all
instructors, who can learn more by visiting:
www.engineering-education.com/CATS.
AcknowledgementsÐSupport by the National Science Foundation under grant REC-0440295 and by the Department of
Mechanical Engineering at Carnegie Mellon University is
gratefully acknowledged.
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Paul S. Steif received undergraduate and graduate degrees from Brown University and
Harvard University in engineering mechanics. He is currently Professor of Mechanical
Engineering at Carnegie Mellon University. He has been active as a teacher and researcher
in the field of engineering mechanics, for which he has received a number of awards. Dr.
Steif is currently involved in research to study student learning in basic engineering subjects,
to measure student conceptual progress, and to construct educational materials that
facilitate learning. Many of these developments have reached an international audience,
including educational software that is published with widely selling textbooks.
Mary Hansen received M.S. and Ph.D. degrees in Research Methodology from the
University of Pittsburgh. She is currently an Assistant Professor in the School of Education
and Social Sciences at Robert Morris University. Her research interests include educational
measurement and assessment.